Nonlinear spectral problems in electromagnetics: asymptotics and inversion.

尽管在计算速度方面取得了非凡的进步，但电磁学中的3D成像问题的分辨率仍然在从生物医学成像到地球物理勘探和材料缺陷的无损检测的广泛应用中构成了重大挑战。例如，通过从钻孔中的探头发送电磁源并阅读结果电位，人们原则上可以推断出石油矿床的位置，石油矿床的电磁特性与周围的地面不同。然而，对于大规模的非线性问题，蛮力计算方法往往是繁琐的，甚至是棘手的。在拟议的研究中，将开发数学方法，其中我们发现数据对电磁参数的依赖性的显式公式，从而导致快速算法微调到特定的几何形状，并具有明确的物理解释。几名学生将在项目过程中接受培训，包括一名全职博士生。该项目还将为本科生提供花六个月的时间深入研究一个研究问题的机会，并可能增加他们在数学或相关领域攻读更高学位的可能性。该项目的目标是通过开发基于材料性质扰动的新渐近公式来解决电磁学中的几个非线性谱问题和相关逆问题。该项目将包括五个主要组成部分：（一）线性和非线性小体积电磁散射体共振值的渐近公式。渐近性将基于简化的极限问题的扰动，其解很容易计算。(ii)在薄的高对比度材料存在下电场和共振的降维公式。这个公式将提供一个高对比度的玻恩近似，我们可以反演。(iii)不同材料特性透射本征值扰动的显式计算。这将包括材料中的小的不均匀性、空隙或孔洞以及周期性的微观结构。这些公式将揭示从远场阅读透射本征值可以恢复哪些材料特性。(iv)超声调制光学层析成像是一种新型的混合成像模式，它为光学参数的恢复提供了非线性内部数据。(v)非线性弦的逆谱方法，德雷克塞尔大学本科生的一个项目。材料扰动渐近性在非线性谱问题中的应用是所有这些项目中的统一主题。

Despite extraordinary advances in computing speed, the resolution of 3D imaging problems in electromagnetics continues to pose significant challenges in a wide range of applications, from biomedical imaging to geophysical exploration and non-destructive detection of material defects. By sending electromagnetic sources from probes in boreholes and reading back resulting potentials, for instance, one can in principle infer the location of oil deposits, which have different electromagnetic properties than the surrounding ground. Brute-force computational approaches, however, are often cumbersome or even intractable for large scale non-linear problems. In the proposed research, mathematical methods will be developed where we find explicit formulas for the dependence of the data on electromagnetic parameters, leading to fast algorithms fine-tuned to specific geometries and having clear physical interpretations. Several students will be trained in the course of the project, including a full time PhD student. The project will also provide undergraduate students the opportunity to spend six months delving deeply into a research problem, and may increase the likelihood they will pursue a higher degree in mathematics or a related field.The objective of this proposal is to solve several nonlinear spectral problems and related inverse problems in electromagnetics by developing novel asymptotic formulas based on material property perturbations. The project will involve five main components: (i) Asymptotic formulas for resonance values in the presence of linear and nonlinear small volume electromagnetic scatters. The asymptotics will be based on perturbations of simplified limiting problems whose solutions are easily computed. (ii) A dimensionally reduced formulation for the electric field and resonances in the presence of thin high contrast materials. This formulation will provide a high contrast Born approximation which we can invert. (iii) Explicit calculations of the perturbations of transmission eigenvalues for varying material properties. This will include small inhomogeneities, voids, or holes in the materials, and periodic microstructures. The formulas will reveal what material properties can be recovered from reading transmission eigenvalues in the far field. (iv) The development of a fast nonlinear inversion technique and a theory of uniqueness and stability in ultrasound modulated optical tomography, a novel hybrid imaging modality which provides nonlinear internal data for the recovery of optical parameters. (v) An inverse spectral method for nonlinear strings, a project for Drexel undergraduate students. The application of material perturbation asymptotics to nonlinear spectral problems is a unifying theme in all of these projects.